Flexagons are a favorite mathematical craft. The original flexagon—the trihexaflexagon—appears to be a two-sided hexagon, but by performing a pinch flex move, the hexagon opens from the center displaying a hidden face. Continuing to pinch flex the shape and we see that the flexagon has three sides. That is not quite true, however. While it…

# Category: Math

## Preorders and Finite Topological Spaces

Today I tweeted that I had asked my topology students to find all of the different topologies of a two-point set and a three-point set. It turns out that there are three of the former and nine of the latter. (The sequence of the number of topologies for an n-point set begins 1 (for ),…

## My Two-Day Crash Course in PGFPlots and TikZ

I will be teaching multivariable calculus in the fall. During the semester, I’ll have to make numerous figures in two-and three-dimensional space for exams and handouts. One of the things I wanted to do this summer was to learn how to use TikZ to create graphs and other graphics in my LaTeX documents. After a…

## Essential Trigonometry for Calculus

My son is taking a calculus course in high school this year. While talking to him about his homework, I have come to realize that his knowledge of trigonometry is pretty weak. He said that they were supposed learn trigonometry last spring after the COVID-19 pandemic hit and school switched to fully online. Needless to…

## Puzzle: Let’s (Un)Do the Twist!

Today, while walking our dog, I had an idea for the puzzle shown below. Here is a printable pdf. I hope you enjoy it. If you would like a hint as to how to solve the puzzle, read about this puzzle from the great puzzle master Sam Loyd; it was the inspiration for my puzzle.

## Mathematicians’ Phone Passcodes

A character in a novel I was reading used the passcode 1729 for his house’s security system. He did so because of the famous Hardy-Ramanujan anecdote about the number. That got me to thinking. What would mathematicians of the past have used for their passcodes? I tweeted some ideas and got some great responses with…

## Make a Real Projective Plane (Boy’s Surface) out of Paper

I am teaching an undergraduate course in topology. We are now looking at what we get if we take a square and glue the sides together. (These are called identification spaces.) We are assuming that our spaces are made out of very stretchy rubber. So, if the space begins as a square, we could, for instance,…

## The Magnificent Möbius Band

As I write this blog post, we are all either struggling with the impact of the COVID-19 virus or waiting nervously as cases start to rise in our area. I am currently teaching remotely. My college students are scattered around the globe, and we are interacting through various online methods. This semester I am teaching…

## How to Present a Mathematical Proof or Problem

There are many useful websites containing advice on how to give a good mathematics presentation (such as those listed here). But these are written for scholars who are giving lectures on their research. They focus on organizing the talk, putting the research in context, deciding what to include or not include, designing slides, pacing and…

## Homemade Klein Bottle

One of my favorite items in my collection of mathematical objects is a glass Klein bottle made by Cliff Stoll. So I was excited when—a few years ago—I saw that it was possible to make a glass-looking Klein bottle out of a Method soap bottle. Eventually, we bought some of this soap and then used…